Download and ferrimagnetic materials based on the modified steinmetz equation

Calculation of Losses in Ferro- and Ferrimagnetic Materials Based on
the Modified Steinmetz Equation
J. Reinert
A. Brockmeyer
1
Institute for Power Electronics and Electrical Drives
Aachen University of Technology
Jagerstr. 17- 19
D-52066 Aachen, Germany
2Siemens AG
Transportation Systems, VT 872
P.O. Box 32 40
D-9 1052 Erlangen
-
Abstract This paper discusses the influence of nonsinusoidal flux-waveforms on the remagnetization losses
in ferro- and ferrimagnetic materials of inductors, transformers and electrical machines used in power electronic
applications. The non-sinusoidal changes of flux originate
from driving these devices by non-sinusoidal voltages and
currents at different switching frequencies. A detailed
examination of a dynamic hysteresis model shows that the
physical origin of losses in magnetic material is the average remagnetization velocity rather than the remagnetization frequency. This principle leads to a modification of
the most common calculation rule for magnetic core
losses, i.e., to the “Modified Steinmetz Equation” (MSE).
In the MSE the remagnetization frequency is replaced by
an equivalent frequency which is calculated from the
average remagnetization velocity. This approach allows,
for the first time, to calculate the losses in the time domain for arbitrary waveforms of flux while using the
available set of parameters of the classical Steinmetz
equation. DC-premagnetization of the material, having a
substantial influence on the losses, can also be included.
Extensive measurements verify the Modified Steinmetz
Equation presented in this paper.
I.
R.W. De Doncker
INTRODUCTION
An exact prediction of the remagnetization losses of ferroor ferrimagnetic material is critical for the design of inductors, transformers and electrical machines. In power electronic applications this task is difficult because in most
applications the magnetic material is exposed to nonsinusoidal flux waveforms. As an example, Fig. 1 shows the
primary and secondary transformer currents of a flyback
converter as a function of time.
A
In another example, Fig. 2 illustrates the phase current and
the corresponding flux linkage in the stator pole of a switched
reluctance (SR) machine. Clearly, both examples show that
the remagnetization processes are non-sinusoidal.
Fig. 2: Phase current and flux linkage of a SR machine in
chopping mode
Although magnetic materials have been subjected to research for over a century, the exceptional conditions and
requirements in power electronic circuits have not been taken
into detailed consideration. The reason for this might be that
most design rules and loss formulas for magnetic materials
were formulated a long time ago for sinusoidal processes.
When applied to inductors, transformers and machines that
are exposed to high induction levels and switching fkequencies, they loose their validity
Therefore, this paper discusses the physical justification
for representing the properties of magnetic components in the
frequency domain.
11. PHYSICAL ORIGIN OF LOSSES
The most common concept for calculating remagnetization
losses is the addition of two separate terms, i.e., the so-called
hysteresis losses and the so-called eddy current losses.
Hence, it was assumed that two separate physical effects are
contributing to the remagnetization losses. Although many
specialists from material sciences and physics have contradicted this hypothesis, most engineers and technicians still
believe this loss separation is correct.
Fig. 1: Idealized transformer currents in the windings of a
flyback converter
0-7803-5589-X/99/$10.000 1999 IEEE
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To shed some light on this topic, a brief overview over the
theory of magnetization and magnetization losses that has
been discussed in literature is given below.
A very commendable introduction to magnetic materials
has been given by Fish [15]. Serious discussions of magnetization theory and the reasons for remagnetization losses can
also be found in the work of Bertotti [16],[17], Graham [l]
and Becker [ 181.
It is known that the magnetization in ferro- and ferrimagnetic materials is not uniform. As shown in Fig. 3, the internal structure of the material can be subdivided into saturated
domains which differ from each other by the orientation of
their magnetization vectors. The magnetic domains are separated from each other by domain walls. A change in the
global magnetization of the material can only be achieved by
movement of these domain walls.
Domain walls
Direction of magnetizationin saturated domains:
Direction of domain wall movement: +
The detailed knowledge about the origin of remagnetization losses does not provide a practical means of calculating
losses. In general, the rather chaotic time- and space- distribution of the magnetization changes is unknown and cannot
be described exactly.
To work around this lack of a microscopic physical remagnetization model, several macroscopic and empirical approaches have been formulated in the past. They can be
subdivided into hysteresis models, empirical equations and
the loss separation approach.
A.
Hysteresis Models
The vast number of hysteresis models can be separated into
two branches. One part is based on the Jiles-Atherton model
and the other part traces back to Preisach’s work.
The Jiles-Atherton model [ 191 is based on a macroscopic
energy calculation. It consists of a differential equation that
describes the static behavior of ferro- and ferrimagnetic behavior. An iterative procedure has to be used to estimate the
parameters of the model. The model can be extended to dynamic calculations [20] which increase the number of required parameters to seven. Additional parameters have to be
used to describe the temperature behavicr and to calculate
minor hysteresis loops. Although this model leads to a better
understanding of the remagnetization process, it is of limited
practical use.
n
Domains
111. CONVENTIONAL CALCULATION
++++ 646s
Fig. 3: Change in a domain structure
This means that the magnetization changes in a highly localized way, rather than uniformly through the material. The
magnetization change is discrete in terms of space.
Impurities and imperfections inside the material hinder the
domain wall motion and cause rapid movements of the domain walls, the so-called Barkhausen-Jumps.
Therefore, the movement of the domain walls is not regular. The local velocity of the walls is not equal to the change
of rate of the external field. This means that the magnetization change is discrete in terms of time.
If the change of magnetization is discrete in terms of space
and in terms of time there have to be rapid local changes of
magnetization, even if the external field changes with an
infinitesimal low rate, i.e., the quasi-static case. Associated
with magnetization changes are local energy losses caused by
eddy currents and by spin-relaxation. These losses are determined by the local- and time-distribution of the changes.
Consequently, there is no physical difference between
“hysteresis” losses and “eddy current” losses. As Graham
pointed out there is no physical distinction to be made between the static losses and the dynamic losses [l]. There is
only one physical origin of remagnetization losses, namely,
the damping of domain wall movement by eddy currents and
spin-relaxation.
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Preisach’s model as described e.g. by Hui [23] introduces a
statistical approach for the description of the time- and space
distribution of domain-wall movement. A weight function
represents the material characteristics. The classical model
exhibits two major drawbacks - the limited congruency of
minor loops and the static character. The model can be extended to dynamic effects but the identification problem
connected with the weight functions results in a tremendous
experimental effort that is not justified by the incremental
increase of accuracy.
B.
Empirical Equations
One well-known empirical equation to calculate remagnetization losses traces back to the original work of Steinmetz
more than a century ago and is formulated by means of an
empirical equation [4]:
It states that the power losses py per volume are dependent
on exponential functions of the remagnetization frequency f
and the peak induction B , using three empirical parameters
C
,,
a, p. Both exponents are non-integer numbers, i.e.,
l < 6 3 and 2<p<3. The appearance of the remagnetization
frequency f in this equation has to be explained by the empirical character of the studies made by Steinmetz a century
ago. The equation and the corresponding set of parameters is
only valid for sinusoidal remagnetization, which is a major
A
drawback for the implementation in power electronic applications.
Gradzki [21] and Severns [22] try to overcome this problem by using a Fourier expansion of the arbitrary nonsinusoidal waveforms. Equation (1) is then applied to each
single Fourier component. Finally, the individual losses of
the fundamental and all harmonics are superimposed and
summarized to calculate the total losses. The fact that the
induction exponent p of the equation has a typical value of
p = 2.5 indicates that there is an extremely non-linear relation
between losses and peak-induction. The method of superposition is mathematically only applicable for linear systems. In
case of non-linear magnetic materials its application is not
valid and the results of this procedure are invalid [7],[8],[9].
For ferromagnetic materials, that are normally used in form
of sheets, laminations or tapes with a fixed thickness, the
losses are specified by the manufacturers as a function of
material, quality and sheet thickness. Equation (1) is used to
extract the parameters from these specifications. This finally
leads to a different set of parameters for each individual material and sheet thickness. Consequently, (1) gives the total
remagnetization losses including static and dynamic eddycurrent losses.
In case of ferrites, losses are specified depending on the
material grade. The dependence on geometry is usually neglected in these specifications. Therefore, it is necessary to
introduce an additional term into the loss equation that accounts for geometric effects:
not influenced by the remagnetization waveform. In this case
Maxwell's theory can be applied and the classic eddy current
calculation finally leads to several form-factors for typical
non-sinusoidal waveforms. These form-factors have the same
poor accuracy as the loss-separation approach.
IV. THENOVELMSE APPROACH
The empirical Steinmetz equation (1) has proven to be the
most useful tool for the calculation of remagnetization losses.
It requires only three parameters which are usually published
by the manufacturer. For sinusoidal flux-waveforms it provides a high accuracy and is quite simple to use.
Therefore, it is desirable to extend this equation to nonsinusoidal problems. This can be done with the help of the
physical understanding taken from the development of dynamic hysteresis models. It has been shown that the macroscopic remagnetization velocity dWdt is directly related to
the core losses [20]. Therefore, the task is quite simple: the
empirical loss parameter frequencyfof (1) has to be replaced
by the physical loss parameter dWdr, which is proportional
to the rate-of-change of the induction dB/dt.
As a first step, the induction change-rate dB/dt is averaged
over a complete remagnetization cycle, thus from maximum
induction B,, down to its minimum Bmi,and back:
1 dB
B=-g-dB,
AB dt
This integral can be transformed:
B = - Jl( $T)
The parameter C, of this equation is dependent on the
cross-section and the conductivity of the core. For medium
frequencies below 100 kHz, the conductivity of ferrites is
typically very low, which means that the geometrical influence on the total losses can be neglected. However, above
100 kHz ferrite may be subjected to dispersion that leads to a
significant increase in conductivity.
,
C.
Loss-Separation Approach
AB= B,, - Bmin
2
dt
The second step consists of finding a relationship between
the remagnetization frequency f and the averaged remagnetization velocity B . It has been shown by Diirbaum [lo] that
(4) can be normalized with respect to a sinusoidal case. From
the averaged remagnetization velocity an equivalent frequencyf,, can be calculated using the normalization constant
2 f ABn2:
The third loss calculation method traces back to Jordan [3]
and separates the total losses P, in two parts, i.e., the static
hysteresis loss Ph and the dynamic eddy-current loss P,.
Pf= Ph iP,
*
(4)
AB0
It has already been shown that this approach lacks theoretical justification. But even the practical use is limited because
of the fact that the results are inaccurate. Many papers report
that calculation errors between 200% and 2000% can occur.
Therefore a third artificial loss component is introduced, the
so called "eddy-current anomaly loss" P,.
Only the eddy current loss P, of the three components can
be calculated. The hysteresis loss and the anomalous losses
have to be determined experimentally.
Non-sinusoidal remagnetization can be taken into account
on the assumption that hysteresis and anomalous losses are
(5)
Similar to the empirical formula of Steinmetz the specific
energy loss w, of every remagnetization cycle can now be
determined using this equivalent frequency:
w, = e, fL.,"-'2
If the remagnetization is repeated with the period Tr = 1 /fr
the power losses are:
(7)
This Modified Steinmetz Equation (MSE) describes the
physical origin of the losses and gives the opportunity to
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B/Bo
calculate the core losses in the time domain for arbitrary
shapes of induction. Compared to the original Steinmetz
equation, no additional parameters are needed.
V. EXPERMENTAL
VERIFICATION
To validate the MSE, an experimental setup according to
the European Standard CECC 25 300 and CECC 25 000, as
shown in Fig. 4, is used. This setup is chosen because of its
high accuracy even for non-linear materials and because it
provides more information about the core material than just
the core losses. According to Carsten [l 11 it is the only core
loss measurement technique without technical disadvantages.
Fig. 5: Triangular remagnetization with varying delay
time, Bo = 200 mT, T'= 20 kHz, U = 100°C
10 0000
1 0000
PvMl
0 1000
Fig. 4: Measurement setup
The device under test (DUT) carries three windings, nl. a
primary AC-winding, a secondary sense winding and a third
winding to introduce a DC-premagnetization via a DC
source. Via a special low-inductive shunt R the primary
winding is connected to an AC-power amplifier, producing
arbitrary remagnetization cycles. The induced voltage at the
DUT is measured by the secondary sense winding. This induced voltage and the voltage drop over the shunt resistor are
sampled by a LeCroy digital storage oscilloscope with
2,5 GS/s and a bandwidth of 300 MHz. The sampled waveforms are transferred to a PC, used to calculate the magnetic
field and the flux density in the core. The core losses can then
directly be determined by the surface of the hysteresis loop.
The phase angle between primary current and secondary
voltage of the DUT is related directly to the core losses. For
low-loss components it differs only slightly from 90". Hence,
the error is typically introduced by the current measurement
device. Therefore, it is crucial important to use high-precision
low-inductive shunts.
A.
0.0100
1000
io000
trmz
100000
Fig. 6: Comparison between calculation and measurement
for triangular remagnetization
Thirdly, the calculation of the core losses is performed by
the Fourier series approach. Fig. 6 shows that the results of
the MSE are in very good agreement with the experimental
results. It also shows that the use of the original Steinmetz
equation, which is only valid for sinusoidal remagnetization,
is actually more accurate than the calculation by the Fourier
series approach.
For the next experiment, the duty cycle of the flux waveform is varied, as shown in Fig. 7.
Measurement Results - Ferrimagnetic
The first experiment is performed with an E42/42/15 Philips 3C85 ferrite core, which is exposed to constant triangular
remagnetization cycles, as is shown in Fig. 5 . In Fig. 6, the
measured core losses are compared to different calculations.
Firstly, the remagnetization losses are calculated from the
original Steinmetz equation (1). Secondly, the modified
equation (7) is used. In this case it simplifies into:
p=C
m
-(--)
1 2 4
a-'
0
TI8
TI4
3Ti0
TI2
5Ti8
3Ti4
?TI8
T
Fig. 7: Remagnetization with different duty cycles
B ~ 2 2 0 m Tl/T=20kHz,
,
u=10OoC
Bop
Fig. 8 shows a comparison of the core losses calculated
from (1) and from (7) with the experimental results. It indicates that the measured core losses increase significantly with
increasing duty cycle. This behavior is also represented by
T, n 2 T
2090
results derived fiom the modified core loss equation, but can
not be predicted with the conventional equation.
27CQ
25w
2300
PlmW
2100
1wx)
can be used to find the flux waveforms in the different parts
of the machine. For the piecewise linear waveforms (see
Fig. lo), equation(7) is used to calculate the losses in the
poles and the yoke sections [13]. Due to the non-uniform flux
distribution and the saturation effects always present in SR
machines, a precise calculation of the iron losses is extremely
dificult. Experiments with the 4-phase machine have shown,
that the error of the iron-loss calculation is smaller than 10%
for .all operating conditions [12]. This accuracy can not be
achieved with any other previously derived method.
&ator pole flux
1700
n
.Stator yoke fliw. (1)
Fig. 8: Comparison between measurement and calculation
as a function of duty cycle.
n
B.
.
2n @
I
Measurement Results - Ferromagnetic
+Stator yoke flux (2)
The first experiment with ferromagnetic material is performed with a Surahammars Bruk CK27 material with a
lamination thickness of 0.35 mm, which is used in a 4-phaseY
30 kW switched reluctance (SR) machine [12]. To be able to
test the influence of different remagnetizations, independently of the lamination geometry, a toroidal core of the same
lamination material was used as the DUT in the setup of
Fig. 4 for the first tests. Subjecting the toroidal core to an
alternating block voltage (duty cycle=SO%, Tr=T) and measuring the losses, gives the results shown in Fig. 9. Again, the
experimental results are compared to the calculated losses
from (1) and (7).
Fig. 10: Flux waveforms in different core parts of a 4phase SR machine in single pulse operation
C.
0
1000
ZOO0
3000
4000
SO00
GOO0
f.(Hzjooo
Fig. 9: Comparison between calculation an measurement
for triangular remagnetization (T, =T)
Increasing the period of the cycle, i.e. T,T (see Fig. S),
leads to an even larger error of the calculation with (1).
In single pulse operation, a SR-drive is operated with a
block voltage scheme, leading to triangular remagnetization
in the poles. From this, the waveforms in the different yoke
sections can be obtained. As an example the flux waveforms
for the 4-phase machine in one specific working point are
shown in Fig. 10.
To calculate the entire iron losses of a SR-machine for
each specific set of control parameters, a simulation program
Infuence of DC-premagnetization
From Fig. 10 it can readily be seen, that during the operation of SR-machines smaller hysteresis sub-loops are encountered under the influence of premagnetization. From
Fig. 1 it is evident, that premagnetization is also commonly
experienced in ferrimagnetic materials used in power electronic applications. Although manufacturers of magnetic
materials never supply data of the influence of premagnetization, it has been shown that it has a major influence on the
losses in both ferromagnetic [ 141 and ferrimagnetic materials
PI,[241.
This influence has been proven by measurement for the
ferromagnetic toroidal core described in section By as shown
in Fig. 11. It can be seen that the losses at a constant ACinduction and frequency increase continuously with the DC
part of the flux density. Similar to the original Steinmetz
equation and any other calculation method, the MSE (7)
cannot incorporate the influence of a premagnetization. In an
empirical approach the loss parameter C,,, in the MSE can be
used to adapt to the influence of premagnetization, as shown
in (9) [71,[121:
209 I
*
cm,new
= G , O l d (1 + K , BLX e Kz
1
(9)
BDc and BACrelate to the constant and the alternating part
of the flux density. The constants K1 and K2, found by measurements at different fkequency and magnetization, describe
the material-dependent influence of premagnetization.
For example, to calculate the iron losses in SR machines,
the magnetization waveforms in the different sections are
divided into their main and sub-loops. Once having determined K1 and K2 the additional losses of the hysteresis loops
under the influence of premagnetization are readily available
for any operating condition.
S. A. Mulder, “Fit formulae for power loss in ferrites and their use in
transformer design”, in PCIM’93 Proc., pp.345-359, ZM Communications, Ntlrnberg, Germany, 6 1993
D. Grtitzer, “Ummagnetisierungsverlusteweichmagnetischer Werkstoffe bei nichtsinusfarmiger Aussteuerung”, Zeitschrift Alr angewandte Physik, 32(3), pp. 241-246, 1971
A. Brockmeyer, “Dimensionierungswerkzeugf i r magnetische
Bauelemente in Stromrichteranwendungen” (in German), PhD. Thesis, Verlag Augustinus Buchhandlung Aachen, ISBN 3-86073-239-0
A. Brockmeyer, M. Albach, T. Dtlrbaum, “Remagnetization losses of
ferrite materials used in power electronic applications”” in PCIM”6
Proc., ZM Communications, Ntlrnberg, Germany, 1996
M. Albach, T. Dllrbaum, A. Brockmeyer, “Calculating core losses in
transformers for arbitrary magnetization currents- a comparison of different approaches”” in PESC’96 Proc., 1996
T. Dllrbaum and M. Albach, ‘.‘Core losses in transformers with an
arbitrary shape of the magnetization current”, in EPE Proc., Vol.1, pp
1.171-1.176,Sevilla (Spain), 1995
B. Carsten, “Why a magnetics designer should measure core loss; with
a survey of loss measurement techniques and a low cost, high accuracy alternative”, in PCIM’95 Proc., pp. 163-179, ZM Communications, Nilrnberg, Germany, 1995
J. Reinert, “Optimierung der Betriebseigenschaften von Antrieben mit
geschalteter Reluktanzmaschine” (in German), PhD. Thesis, Verlag
Augustinus Buchhandlung Aachen, ISBN 3-86073-682-5
0,5--
J. Reinert, R. Inderka, R.W. De Doncker, “A Novel Method for the
Prediction of Losses in Switched Reluctance Machines”, in EPE’97
Proc. Vol. 3, pp 3.608-3.612,Trondheim, 1997
+MSE with Correction
-w
R. M. Bozorth, “Ferromagnetism”, IEEE Press Reprint, Piscataway
NJ, USA, 1993
MSE without Correction
G.E. Fish,”Sofi magnetic materials”, Proc. IEEE, Vol. 78, No. 6, June
1990
G. Bertotti, “General properties of power losses in soft ferromagnetic
materials”, IEEE Trans. Mag., Vol. 24, No. 1, Jan. 1988
VI. CONCLUSIONS
In this paper the physical justification and the experimental
verification of a novel method for the calculation of core
losses for non-sinusoidal induction are presented. This calculation method, called the Modified Steinmetz Equation
(MSE), is particularly useful for the design of magnetic components for power electronic applications and electric machine theory due to its simplicity and because all necessary
parameters are readily available.
G. Bertotti, F. Fiorillo, P. Mazetti, “Basic principles of magnetization
processes and origin of losses in soft magnetic materials”, Journal of
Magnetism and Magnetic Materials 112 (1992) 146-149, North Holland
J. J. Becker, “Magnetization changes and losses in conducting ferromagnetic materials”, Jour. Appl. Phys., Vol. 34, No. 4, April 1963
D. C. Jiles, D. L. Atherton, “Theory of ferromagnetic hysteresis’,,
1986, Journal of Magnetism and Magnetic Materials 61,48-60,North
Holland
A. Brockmeyer, L. Schlllting, “Modelling of dynamic losses in magnetic material”, EPE Proc. 1993, Brighton, U.K.
The tests undertaken for the ferro- and ferrimagnetic material show that the modified equation enables accurate calculation of remagnetization losses for arbitrary flux waveforms.
P. M. Gradzki, M. M. Jovanovic, F. C. Lee, “Computer aided design
for high-frequency power transformers”, in IEEE APEC Proc. 1990,
336-343
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